I went to a private school, and we're considering a totally different type of private school for our daughter (no decisions yet).
Mine was very academic. Exams to get in, regular exams throughout, a lot of emphasis on grades. Very strict on uniform (which was also expensive) and a lot of push towards playing (orchestral) musical instruments and playing traditional sports. It was very disciplined - being caught with cigarettes was expulsion. In high school, obviously, though discipline was very strict at all ages.
I did well in that environment. I was the kind of kid who thrived on structure and there was no classroom disruption. I was always one of the poorest kids because I was on bursary but it didn't hold me back, quite a few of the families were just ordinary folks who really saved to send their kids (though there were a few really posh families too). I did get good grades, but I was naturally academic so I might've done just as well at state school. The rigid rules and old traditions were grating in high school when you want to be independent and individual, and the only subjects offered were quite traditional ones (maths, english, science, languages etc) as opposed to some of the more vocational and unusual ones other state high schools offered. It did not suit everyone, it could definitely be stuffy and only really catered to book-smarts.
One thing in it's credit though is that there was next to zero bullying. Of course not everyone got on, but any sort of bullying behaviour was absolutely not tolerated, it was very safe.
The one we're considering for my daughter is totally different.
Their exam results are no better than the local state schools. It's still very expensive, down to the uniforms. They put a huge focus on kids being well-rounded and finding their strengths. They go hiking and camping (even in primary school) they work on the school farm, and they do a lot of life-skills and physical lessons. It's much smaller and a family sort of environment. I think a lot of the kids who didn't do well in my school might have done great here, because it values social skills and independence as much as (if not more than) exam results. My daughter is so adventurous, I think she would thrive in that environment. I was a shy and nervous child and probably would have hated it.
So it depends a huge amount on whether the school's ethos matches your child's needs.
General points about private schools in general - they often have longer holidays. Good for cheap off-peak holidays, bad for childcare costs. Uniforms, school trips, school buses etc can add a lot of expense. The students tend to come from a wider area, so your child's best friend might end up living a long way away, which gets to be a pain for parties and sleepovers.